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Data MiningCluster Analysis Basic Concepts and

Algorithms

- Lecture Notes for Chapter 8
- Introduction to Data Mining
- by
- Tan, Steinbach, Kumar

10/30/2007 Introduction to Data Mining

1

What is Cluster Analysis?

- Finding groups of objects such that the objects

in a group will be similar (or related) to one

another and different from (or unrelated to) the

objects in other groups

Applications of Cluster Analysis

- Understanding
- Group related documents for browsing, group genes

and proteins that have similar functionality, or

group stocks with similar price fluctuations - Summarization
- Reduce the size of large data sets

Clustering precipitation in Australia

What is not Cluster Analysis?

- Simple segmentation
- Dividing students into different registration

groups alphabetically, by last name - Results of a query
- Groupings are a result of an external

specification - Clustering is a grouping of objects based on the

data - Supervised classification
- Have class label information
- Association Analysis
- Local vs. global connections

Notion of a Cluster can be Ambiguous

Types of Clusterings

- A clustering is a set of clusters
- Important distinction between hierarchical and

partitional sets of clusters - Partitional Clustering
- A division data objects into non-overlapping

subsets (clusters) such that each data object is

in exactly one subset - Hierarchical clustering
- A set of nested clusters organized as a

hierarchical tree

Partitional Clustering

Original Points

Hierarchical Clustering

Traditional Hierarchical Clustering

Traditional Dendrogram

Non-traditional Hierarchical Clustering

Non-traditional Dendrogram

Other Distinctions Between Sets of Clusters

- Exclusive versus non-exclusive
- In non-exclusive clusterings, points may belong

to multiple clusters. - Can represent multiple classes or border points
- Fuzzy versus non-fuzzy
- In fuzzy clustering, a point belongs to every

cluster with some weight between 0 and 1 - Weights must sum to 1
- Probabilistic clustering has similar

characteristics - Partial versus complete
- In some cases, we only want to cluster some of

the data - Heterogeneous versus homogeneous
- Clusters of widely different sizes, shapes, and

densities

Types of Clusters

- Well-separated clusters
- Center-based clusters
- Contiguous clusters
- Density-based clusters
- Property or Conceptual
- Described by an Objective Function

Types of Clusters Well-Separated

- Well-Separated Clusters
- A cluster is a set of points such that any point

in a cluster is closer (or more similar) to every

other point in the cluster than to any point not

in the cluster.

3 well-separated clusters

Types of Clusters Center-Based

- Center-based
- A cluster is a set of objects such that an

object in a cluster is closer (more similar) to

the center of a cluster, than to the center of

any other cluster - The center of a cluster is often a centroid, the

average of all the points in the cluster, or a

medoid, the most representative point of a

cluster

4 center-based clusters

Types of Clusters Contiguity-Based

- Contiguous Cluster (Nearest neighbor or

Transitive) - A cluster is a set of points such that a point in

a cluster is closer (or more similar) to one or

more other points in the cluster than to any

point not in the cluster.

8 contiguous clusters

Types of Clusters Density-Based

- Density-based
- A cluster is a dense region of points, which is

separated by low-density regions, from other

regions of high density. - Used when the clusters are irregular or

intertwined, and when noise and outliers are

present.

6 density-based clusters

Types of Clusters Conceptual Clusters

- Shared Property or Conceptual Clusters
- Finds clusters that share some common property or

represent a particular concept. - .

2 Overlapping Circles

Types of Clusters Objective Function

- Clusters Defined by an Objective Function
- Finds clusters that minimize or maximize an

objective function. - Enumerate all possible ways of dividing the

points into clusters and evaluate the goodness'

of each potential set of clusters by using the

given objective function. (NP Hard) - Can have global or local objectives.
- Hierarchical clustering algorithms typically

have local objectives - Partitional algorithms typically have global

objectives - A variation of the global objective function

approach is to fit the data to a parameterized

model. - Parameters for the model are determined from the

data. - Mixture models assume that the data is a

mixture' of a number of statistical

distributions.

Map Clustering Problem to a Different Problem

- Map the clustering problem to a different domain

and solve a related problem in that domain - Proximity matrix defines a weighted graph, where

the nodes are the points being clustered, and the

weighted edges represent the proximities between

points - Clustering is equivalent to breaking the graph

into connected components, one for each cluster. - Want to minimize the edge weight between clusters

and maximize the edge weight within clusters

Characteristics of the Input Data Are Important

- Type of proximity or density measure
- Central to clustering
- Sparseness
- Dictates type of similarity
- Adds to efficiency
- Attribute type
- Dictates type of similarity
- Type of Data
- Dictates type of similarity
- Other characteristics, e.g., autocorrelation
- Dimensionality
- Noise and Outliers
- Type of Distribution

Clustering Algorithms

- K-means and its variants
- Hierarchical clustering
- Density-based clustering

K-means Clustering

- Partitional clustering approach
- Number of clusters, K, must be specified
- Each cluster is associated with a centroid

(center point) - Each point is assigned to the cluster with the

closest centroid - The basic algorithm is very simple

Example of K-means Clustering

Example of K-means Clustering

K-means Clustering Details

- Initial centroids are often chosen randomly.
- Clusters produced vary from one run to another.
- The centroid is (typically) the mean of the

points in the cluster. - Closeness is measured by Euclidean distance,

cosine similarity, correlation, etc. - K-means will converge for common similarity

measures mentioned above. - Most of the convergence happens in the first few

iterations. - Often the stopping condition is changed to Until

relatively few points change clusters - Complexity is O( n K I d )
- n number of points, K number of clusters, I

number of iterations, d number of attributes

Evaluating K-means Clusters

- Most common measure is Sum of Squared Error (SSE)
- For each point, the error is the distance to the

nearest cluster - To get SSE, we square these errors and sum them.
- x is a data point in cluster Ci and mi is the

representative point for cluster Ci - can show that mi corresponds to the center

(mean) of the cluster - Given two sets of clusters, we prefer the one

with the smallest error - One easy way to reduce SSE is to increase K, the

number of clusters - A good clustering with smaller K can have a lower

SSE than a poor clustering with higher K

Two different K-means Clusterings

Original Points

Sub-optimal Clustering

Optimal Clustering

Limitations of K-means

- K-means has problems when clusters are of

differing - Sizes
- Densities
- Non-globular shapes
- K-means has problems when the data contains

outliers.

Limitations of K-means Differing Sizes

K-means (3 Clusters)

Original Points

Limitations of K-means Differing Density

K-means (3 Clusters)

Original Points

Limitations of K-means Non-globular Shapes

Original Points

K-means (2 Clusters)

Overcoming K-means Limitations

Original Points K-means Clusters

One solution is to use many clusters. Find parts

of clusters, but need to put together.

Overcoming K-means Limitations

Original Points K-means Clusters

Overcoming K-means Limitations

Original Points K-means Clusters

Importance of Choosing Initial Centroids

Importance of Choosing Initial Centroids

Importance of Choosing Initial Centroids

Importance of Choosing Initial Centroids

Problems with Selecting Initial Points

- If there are K real clusters then the chance of

selecting one centroid from each cluster is

small. - Chance is relatively small when K is large
- If clusters are the same size, n, then
- For example, if K 10, then probability

10!/1010 0.00036 - Sometimes the initial centroids will readjust

themselves in right way, and sometimes they

dont - Consider an example of five pairs of clusters

10 Clusters Example

Starting with two initial centroids in one

cluster of each pair of clusters

10 Clusters Example

Starting with two initial centroids in one

cluster of each pair of clusters

10 Clusters Example

Starting with some pairs of clusters having three

initial centroids, while other have only one.

10 Clusters Example

Starting with some pairs of clusters having three

initial centroids, while other have only one.

Solutions to Initial Centroids Problem

- Multiple runs
- Helps, but probability is not on your side
- Sample and use hierarchical clustering to

determine initial centroids - Select more than k initial centroids and then

select among these initial centroids - Select most widely separated
- Postprocessing
- Bisecting K-means
- Not as susceptible to initialization issues

Empty Clusters

- K-means can yield empty clusters

Empty Cluster

Handling Empty Clusters

- Basic K-means algorithm can yield empty clusters
- Several strategies
- Choose the point that contributes most to SSE
- Choose a point from the cluster with the highest

SSE - If there are several empty clusters, the above

can be repeated several times.

Updating Centers Incrementally

- In the basic K-means algorithm, centroids are

updated after all points are assigned to a

centroid - An alternative is to update the centroids after

each assignment (incremental approach) - Each assignment updates zero or two centroids
- More expensive
- Introduces an order dependency
- Never get an empty cluster
- Can use weights to change the impact

Pre-processing and Post-processing

- Pre-processing
- Normalize the data
- Eliminate outliers
- Post-processing
- Eliminate small clusters that may represent

outliers - Split loose clusters, i.e., clusters with

relatively high SSE - Merge clusters that are close and that have

relatively low SSE - Can use these steps during the clustering process
- ISODATA

Bisecting K-means

- Bisecting K-means algorithm
- Variant of K-means that can produce a partitional

or a hierarchical clustering

Bisecting K-means Example

Hierarchical Clustering

- Produces a set of nested clusters organized as a

hierarchical tree - Can be visualized as a dendrogram
- A tree like diagram that records the sequences of

merges or splits

Strengths of Hierarchical Clustering

- Do not have to assume any particular number of

clusters - Any desired number of clusters can be obtained by

cutting the dendrogram at the proper level - They may correspond to meaningful taxonomies
- Example in biological sciences (e.g., animal

kingdom, phylogeny reconstruction, )

Hierarchical Clustering

- Two main types of hierarchical clustering
- Agglomerative
- Start with the points as individual clusters
- At each step, merge the closest pair of clusters

until only one cluster (or k clusters) left - Divisive
- Start with one, all-inclusive cluster
- At each step, split a cluster until each cluster

contains a point (or there are k clusters) - Traditional hierarchical algorithms use a

similarity or distance matrix - Merge or split one cluster at a time

Agglomerative Clustering Algorithm

- More popular hierarchical clustering technique
- Basic algorithm is straightforward
- Compute the proximity matrix
- Let each data point be a cluster
- Repeat
- Merge the two closest clusters
- Update the proximity matrix
- Until only a single cluster remains
- Key operation is the computation of the proximity

of two clusters - Different approaches to defining the distance

between clusters distinguish the different

algorithms

Starting Situation

- Start with clusters of individual points and a

proximity matrix

Proximity Matrix

Intermediate Situation

- After some merging steps, we have some clusters

C3

C4

Proximity Matrix

C1

C5

C2

Intermediate Situation

- We want to merge the two closest clusters (C2 and

C5) and update the proximity matrix.

C3

C4

Proximity Matrix

C1

C5

C2

After Merging

- The question is How do we update the proximity

matrix?

C2 U C5

C1

C3

C4

?

C1

? ? ? ?

C2 U C5

C3

?

C3

C4

?

C4

Proximity Matrix

C1

C2 U C5

How to Define Inter-Cluster Distance

Similarity?

- MIN
- MAX
- Group Average
- Distance Between Centroids
- Other methods driven by an objective function
- Wards Method uses squared error

Proximity Matrix

How to Define Inter-Cluster Similarity

- MIN
- MAX
- Group Average
- Distance Between Centroids
- Other methods driven by an objective function
- Wards Method uses squared error

Proximity Matrix

How to Define Inter-Cluster Similarity

- MIN
- MAX
- Group Average
- Distance Between Centroids
- Other methods driven by an objective function
- Wards Method uses squared error

Proximity Matrix

How to Define Inter-Cluster Similarity

- MIN
- MAX
- Group Average
- Distance Between Centroids
- Other methods driven by an objective function
- Wards Method uses squared error

Proximity Matrix

How to Define Inter-Cluster Similarity

?

?

- MIN
- MAX
- Group Average
- Distance Between Centroids
- Other methods driven by an objective function
- Wards Method uses squared error

Proximity Matrix

MIN or Single Link

- Proximity of two clusters is based on the two

closest points in the different clusters - Determined by one pair of points, i.e., by one

link in the proximity graph - Example

Distance Matrix

Hierarchical Clustering MIN

Nested Clusters

Dendrogram

Strength of MIN

Original Points

Six Clusters

- Can handle non-elliptical shapes

Limitations of MIN

Two Clusters

Original Points

- Sensitive to noise and outliers

Three Clusters

MAX or Complete Linkage

- Proximity of two clusters is based on the two

most distant points in the different clusters - Determined by all pairs of points in the two

clusters

Distance Matrix

Hierarchical Clustering MAX

Nested Clusters

Dendrogram

Strength of MAX

Original Points

Two Clusters

- Less susceptible to noise and outliers

Limitations of MAX

Original Points

Two Clusters

- Tends to break large clusters
- Biased towards globular clusters

Group Average

- Proximity of two clusters is the average of

pairwise proximity between points in the two

clusters. - Need to use average connectivity for scalability

since total proximity favors large clusters

Distance Matrix

Hierarchical Clustering Group Average

Nested Clusters

Dendrogram

Hierarchical Clustering Group Average

- Compromise between Single and Complete Link
- Strengths
- Less susceptible to noise and outliers
- Limitations
- Biased towards globular clusters

Cluster Similarity Wards Method

- Similarity of two clusters is based on the

increase in squared error when two clusters are

merged - Similar to group average if distance between

points is distance squared - Less susceptible to noise and outliers
- Biased towards globular clusters
- Hierarchical analogue of K-means
- Can be used to initialize K-means

Hierarchical Clustering Comparison

MIN

MAX

Wards Method

Group Average

MST Divisive Hierarchical Clustering

- Build MST (Minimum Spanning Tree)
- Start with a tree that consists of any point
- In successive steps, look for the closest pair of

points (p, q) such that one point (p) is in the

current tree but the other (q) is not - Add q to the tree and put an edge between p and q

MST Divisive Hierarchical Clustering

- Use MST for constructing hierarchy of clusters

Hierarchical Clustering Time and Space

requirements

- O(N2) space since it uses the proximity matrix.
- N is the number of points.
- O(N3) time in many cases
- There are N steps and at each step the size, N2,

proximity matrix must be updated and searched - Complexity can be reduced to O(N2 log(N) ) time

with some cleverness

Hierarchical Clustering Problems and Limitations

- Once a decision is made to combine two clusters,

it cannot be undone - No objective function is directly minimized
- Different schemes have problems with one or more

of the following - Sensitivity to noise and outliers
- Difficulty handling different sized clusters and

convex shapes - Breaking large clusters

DBSCAN

- DBSCAN is a density-based algorithm.
- Density number of points within a specified

radius (Eps) - A point is a core point if it has more than a

specified number of points (MinPts) within Eps - These are points that are at the interior of a

cluster - A border point has fewer than MinPts within Eps,

but is in the neighborhood of a core point - A noise point is any point that is not a core

point or a border point.

DBSCAN Core, Border, and Noise Points

DBSCAN Algorithm

- Eliminate noise points
- Perform clustering on the remaining points

DBSCAN Core, Border and Noise Points

Original Points

Point types core, border and noise

Eps 10, MinPts 4

When DBSCAN Works Well

Original Points

- Resistant to Noise
- Can handle clusters of different shapes and sizes

When DBSCAN Does NOT Work Well

(MinPts4, Eps9.75).

Original Points

- Varying densities
- High-dimensional data

(MinPts4, Eps9.92)

DBSCAN Determining EPS and MinPts

- Idea is that for points in a cluster, their kth

nearest neighbors are at roughly the same

distance - Noise points have the kth nearest neighbor at

farther distance - So, plot sorted distance of every point to its

kth nearest neighbor

Cluster Validity

- For supervised classification we have a variety

of measures to evaluate how good our model is - Accuracy, precision, recall
- For cluster analysis, the analogous question is

how to evaluate the goodness of the resulting

clusters? - But clusters are in the eye of the beholder!
- Then why do we want to evaluate them?
- To avoid finding patterns in noise
- To compare clustering algorithms
- To compare two sets of clusters
- To compare two clusters

Clusters found in Random Data

Random Points

Different Aspects of Cluster Validation

- Determining the clustering tendency of a set of

data, i.e., distinguishing whether non-random

structure actually exists in the data. - Comparing the results of a cluster analysis to

externally known results, e.g., to externally

given class labels. - Evaluating how well the results of a cluster

analysis fit the data without reference to

external information. - - Use only the data
- Comparing the results of two different sets of

cluster analyses to determine which is better. - Determining the correct number of clusters.
- For 2, 3, and 4, we can further distinguish

whether we want to evaluate the entire clustering

or just individual clusters.

Measures of Cluster Validity

- Numerical measures that are applied to judge

various aspects of cluster validity, are

classified into the following three types. - External Index Used to measure the extent to

which cluster labels match externally supplied

class labels. - Entropy
- Internal Index Used to measure the goodness of

a clustering structure without respect to

external information. - Sum of Squared Error (SSE)
- Relative Index Used to compare two different

clusterings or clusters. - Often an external or internal index is used for

this function, e.g., SSE or entropy - Sometimes these are referred to as criteria

instead of indices - However, sometimes criterion is the general

strategy and index is the numerical measure that

implements the criterion.

Measuring Cluster Validity Via Correlation

- Two matrices
- Proximity Matrix
- Ideal Similarity Matrix
- One row and one column for each data point
- An entry is 1 if the associated pair of points

belong to the same cluster - An entry is 0 if the associated pair of points

belongs to different clusters - Compute the correlation between the two matrices
- Since the matrices are symmetric, only the

correlation between n(n-1) / 2 entries needs to

be calculated. - High correlation indicates that points that

belong to the same cluster are close to each

other. - Not a good measure for some density or contiguity

based clusters.

Measuring Cluster Validity Via Correlation

- Correlation of ideal similarity and proximity

matrices for the K-means clusterings of the

following two data sets.

Corr -0.9235

Corr -0.5810

Using Similarity Matrix for Cluster Validation

- Order the similarity matrix with respect to

cluster labels and inspect visually.

Using Similarity Matrix for Cluster Validation

- Clusters in random data are not so crisp

DBSCAN

Using Similarity Matrix for Cluster Validation

- Clusters in random data are not so crisp

K-means

Using Similarity Matrix for Cluster Validation

- Clusters in random data are not so crisp

Complete Link

Using Similarity Matrix for Cluster Validation

DBSCAN

Internal Measures SSE

- Clusters in more complicated figures arent well

separated - Internal Index Used to measure the goodness of

a clustering structure without respect to

external information - SSE
- SSE is good for comparing two clusterings or two

clusters (average SSE). - Can also be used to estimate the number of

clusters

Internal Measures SSE

- SSE curve for a more complicated data set

SSE of clusters found using K-means

Framework for Cluster Validity

- Need a framework to interpret any measure.
- For example, if our measure of evaluation has the

value, 10, is that good, fair, or poor? - Statistics provide a framework for cluster

validity - The more atypical a clustering result is, the

more likely it represents valid structure in the

data - Can compare the values of an index that result

from random data or clusterings to those of a

clustering result. - If the value of the index is unlikely, then the

cluster results are valid - These approaches are more complicated and harder

to understand. - For comparing the results of two different sets

of cluster analyses, a framework is less

necessary. - However, there is the question of whether the

difference between two index values is

significant

Statistical Framework for SSE

- Example
- Compare SSE of 0.005 against three clusters in

random data - Histogram shows SSE of three clusters in 500 sets

of random data points of size 100 distributed

over the range 0.2 0.8 for x and y values

Statistical Framework for Correlation

- Correlation of ideal similarity and proximity

matrices for the K-means clusterings of the

following two data sets.

Corr -0.9235

Corr -0.5810

Internal Measures Cohesion and Separation

- Cluster Cohesion Measures how closely related

are objects in a cluster - Example SSE
- Cluster Separation Measure how distinct or

well-separated a cluster is from other clusters - Example Squared Error
- Cohesion is measured by the within cluster sum of

squares (SSE) - Separation is measured by the between cluster sum

of squares - Where Ci is the size of cluster i

Internal Measures Cohesion and Separation

- Example SSE
- BSS WSS constant

m

?

?

?

1

2

3

4

5

m1

m2

K1 cluster

K2 clusters

Internal Measures Cohesion and Separation

- A proximity graph based approach can also be used

for cohesion and separation. - Cluster cohesion is the sum of the weight of all

links within a cluster. - Cluster separation is the sum of the weights

between nodes in the cluster and nodes outside

the cluster.

cohesion

separation

Internal Measures Silhouette Coefficient

- Silhouette Coefficient combine ideas of both

cohesion and separation, but for individual

points, as well as clusters and clusterings - For an individual point, i
- Calculate a average distance of i to the points

in its cluster - Calculate b min (average distance of i to

points in another cluster) - The silhouette coefficient for a point is then

given by s (b a) / max(a,b) - Typically between 0 and 1.
- The closer to 1 the better.
- Can calculate the average silhouette coefficient

for a cluster or a clustering

External Measures of Cluster Validity Entropy

and Purity

Final Comment on Cluster Validity

- The validation of clustering structures is

the most difficult and frustrating part of

cluster analysis. - Without a strong effort in this direction,

cluster analysis will remain a black art

accessible only to those true believers who have

experience and great courage. - Algorithms for Clustering Data, Jain and Dubes